Why There is No Memory Burden in Holographic Space-time Models of Black Hole Formation and Evaporation

T. Banks

REVIEW 3 major objections 4 minor 14 references

Why There is No Memory Burden in Holographic Space-time Models of Black Hole Formation and Evaporation

T0 review · 3 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read This paper argues that in Holographic Space-time models the proposed memory-burden effect, which would slow black hole evaporation, is outweighed by the phase space opened up as modular time evolves, so evaporation proceeds at the…

desk verdict Banks's toy model is a plausible counterexample to universal memory burden, but the missing Fermi Golden Rule matrix elements and unfinished HST embedding keep the main conclusion conditional. read the letter →

arxiv 2608.12139 v1 pith:YZEF4ZZT submitted 2026-08-12 hep-th gr-qchep-ph

T. Banks This is my paper · ORCID

classification hep-thgr-qchep-ph PACS 04.70.Dy04.60.-m

keywords memoryburdenblackholeevaporationHolographicSpace-timeFermi'sGoldenRuledetailedbalancehalf-sidedmodularinclusioncausaldiamondsphasespace

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that the 'memory burden' recently proposed as a quantum-information constraint on black hole evaporation does not exist in Holographic Space-time (HST) models of black hole formation and decay. In the author's finite-dimensional toy model, a high-entropy meta-stable state—the model's black hole—does freeze a large number of q-bits while it exists, but as modular time evolves the Hilbert space available to decay products grows so much that the memory burden is more than compensated. The apparent conflict with Fermi's Golden Rule is resolved by how HST defines energy and implements causality through nested, time-dependent unitary embeddings. If correct, the paper removes a proposed reason to expect large deviations from semiclassical Hawking evaporation in astrophysical and cosmological black holes.

What carries the argument

The carrying mechanism is a sequence of unitary embeddings between Hilbert spaces of nested 'causal diamonds,' the finite-dimensional analogues of half-sided modular inclusions in algebraic quantum field theory. Each embedding maps the empty-diamond density matrix with $N$ fermion fields into the next with $N+1$ fields, with geodesic time evolution rescaled by the Milne redshift (a factor of $1/N$ in Planck units), and with an undetermined unitary $U_{\rm out}(N)$ acting on the outside degrees of freedom. Because the Hamiltonians are single-trace operators with 't Hooft scaling, the sum of small block sizes is an approximately conserved quantum number identified with energy; the counting of block diagonalizations then supplies the phase-space enhancement that defeats the memory burden.

What would settle it

Compute, in the toy model, Fermi's Golden Rule decay rate for the N-block bound state into r fragmented blocks at $K\gg N$; if the measured lifetime grows with N rather than following the phase-space-enhanced semiclassical rate, the central claim fails.

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Extended reading notes

Core claim

The central claim is that a meta-stable high-entropy state's decay rate is not suppressed by memory burden, because the entropy lost in freezing q-bits is outweighed by the phase space of fragment configurations accessible in the larger system. Starting from a bound state with N fermion fields in one block, the probability that the equilibrium density matrix has the block split into pieces of sizes $p_i$ is of order $e^{-cpN}$ with $c\sim1$, but once the system has grown to $K\gg N$ fields, the number of ways to arrange the fragments grows like $K!/\prod_i p_i!(K-r)!$, while the intact bound state has only $K-N$ placements. Fermi's Golden Rule therefore favors decay, so the lifetime remains consistent with single-channel semiclassical estimates. The argument works only because HST evolution is a sequence of unitary embeddings—run backward to form the bound state and forward to let it decay—with time-dependent single-trace Hamiltonians implementing causality.

Load-bearing premise

The argument applies to real black holes only if the toy model's nested unitary time evolutions—with the undetermined evolution acting on degrees of freedom outside each causal diamond—faithfully describe black hole formation; the paper itself calls the construction of the corresponding sequence of unitaries the biggest lacuna in the HST formalism.

Editorial extensions

If this is right

Black holes in HST models evaporate at the rate expected from Hawking's semiclassical calculation, with no memory-burden slowdown.
Frozen q-bits in a meta-stable bound state are not, by themselves, an obstruction to decay: the larger Hilbert space reached as modular time evolves restores the phase space.
The lifetime of a high-entropy state depends on the definition of energy and on causal factorization of time evolution, not just on quantum-information counting.
If the conclusion extends to real black holes, observational searches for memory-burden effects in gravitational waves or primordial black holes would lose their theoretical motivation from this mechanism.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial extension: the same phase-space-over-frozen-constraints counting would apply to any finite-dimensional meta-stable system whose decay products can roam over a Hilbert space much larger than the number of constrained q-bits; a numerical simulation of the toy model at $K\gg N$ could test the predicted decay-rate scaling directly.
Editorial extension: if memory burden is model-dependent in this way, constraints derived from bulk effective field theory—which the paper argues cannot account for black hole entropy—may not be universal quantum-information bounds.
Editorial extension: the paper's distinction between stable states, where burden is invisible because decay is absent, and evaporating states, where fragmentation phase space dominates, suggests memory-burden effects may surface mainly in systems whose decay channels are artificially blocked.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this paper does one genuinely new thing—it gives a finite-dimensional toy model where the memory burden mechanism is present but does not slow decay, and argues by phase-space counting that the burden is overwhelmed. That is a real contribution to the memory-burden debate, and the author is unusually honest about the cost: the HST framework is incomplete, and the toy model is not yet a model of real black holes. But the paper's central estimate is not as solid as the prose suggests. The Fermi Golden Rule argument quotes densities of final states but never estimates the off-diagonal matrix elements V. The stress-test note is right on this. Per-channel couplings are switched on 'only a few at a time, at a slower and slower rate as the system grows'; if that rate is exponentially small in N, the polynomial phase-space factor K^r is beside the point. The burden could survive the counting. Also, the counting itself looks partial: it counts placements of blocks among K slots, not the full microstate entropy of the fragments. For a black-hole analogue, the one-block sector has entropy O(N^2); the r-fragment sectors with sum p_i = N have at most O(N log K) states for K ~ N^3, so it is not obvious the final phase space dominates. The conclusion might still be right, but the paper needs to show it.

What is good: the toy model is a clean setup with a well-defined 'energy' (block-size sum), explicit assumptions, and no claim to more than it establishes. The point that a unitary embedding, run backwards and then forwards, naturally produces metastable states whose decay competes with phase-space growth is worth taking seriously. The author flags the biggest gap himself—the sequence of unitaries is 'the biggest lacuna in the HST formalism'—and the Hilbert-bundle section reads as a research program, not a derivation. The citation pattern is heavily self-referential, but that is expected for a framework paper; the engagement with Dvali's papers is direct and fair.

Who this is for: people working on memory burden, PBH constraints, or HST. They will get a clear statement of a possible loophole and a concrete model to argue with. The paper deserves a serious referee. I would recommend conditional acceptance after the author either estimates V or sharply bounds the per-channel suppression, and after clarifying the microstate counting. Fix those and it becomes a solid contribution.

Referee Report

3 major / 4 minor

Summary. The paper argues that memory burden, as proposed by Dvali and collaborators, does not slow black hole evaporation in Holographic Space-Time (HST) models. The argument is developed in a finite-dimensional toy model of fermion fields on an interval, with nested Hilbert spaces connected by unitary embeddings that mimic half-sided modular inclusion. The author defines an approximately conserved quantum number (the sum of block sizes of constrained fermion bilinears) as the analog of energy, and claims that a high-entropy meta-stable state formed by concentrating this energy decays at the rate expected from simple phase-space counting. The second half of the paper sketches a Hilbert bundle over the space of time-like geodesics, using the Quantum Principle of Relativity to restore Poincare covariance, and argues that the toy model can be embedded in a candidate theory of quantum gravity. The conclusion is that memory burden is not a universal obstruction and that HST models match semiclassical evaporation expectations.

Significance. If the central claim were established, it would provide a concrete counterexample to the general memory burden arguments that have recently been applied to primordial black holes and gravitational wave signatures, and it would clarify how HST models avoid the effect. The paper is valuable for making the proposed mechanism explicit and for being unusually candid about the assumptions and open problems in HST, especially the statement that the construction of the modular-embedding unitaries is the biggest lacuna in the formalism. However, the main dynamical estimate is incomplete: the Fermi Golden Rule calculation quotes only the density of final states and never estimates the transition matrix elements, and the embedding of the toy model into a covariant HST framework is explicitly acknowledged to be unproven. The paper therefore establishes at most a conditional claim, not a general theorem about black hole evaporation.

major comments (3)

[Section 2, Fermi Golden Rule paragraph] The central decay estimate is incomplete. The paper concludes that the combinatorial factor K!/[∏ p_i! (K−r)!] 'heavily favors the decay', but Fermi's Golden Rule requires a rate Γ ∝ |V|^2 ρ(E_f), and the off-diagonal matrix elements V are never estimated. The text merely states that off-diagonal elements are turned on a few at a time at a slower and slower rate as the system grows; this is a statement about per-channel suppression. If the per-channel amplitude decays as e^{−cN} or e^{−cNK}, it will overwhelm the polynomial factor K^r, and the toy model would exhibit memory burden. Without an upper or lower bound on these matrix elements, the claimed balance against memory burden is not established even at the toy-model level.
[Section 2, phase-space counting] The phase-space ratio counts only the number of ways to place r blocks among K positions, but the relevant density of states for Fermi's Golden Rule should count all microstates of the larger Hilbert space satisfying the constraints, including the internal states of the fragments and of the original bound state. The one-block sector already contains of order N fermion fields with nontrivial internal structure, and the fragment sectors should be weighted by their own internal densities. If the internal entropy of the r-fragment configurations is much smaller than the entropy of the one-block configuration, the simple combinatorial factor K^r may not dominate. The paper should provide the full density ratio or at least an estimate of the internal-state degeneracies before concluding that detailed balance favors decay.
[Section 2, paragraph on U_out(N), and Section 3] The applicability of the toy model to real black holes depends on the existence of the sequence of unitary embeddings and on a choice of U_out(N) that produces the black-hole-formation initial conditions. The paper itself states, in the bullet list of Section 2, that 'the construction of the corresponding sequence of unitaries is the biggest lacuna in the HST formalism,' and in Section 3 that the Quantum Principle of Relativity argument 'does not guarantee that the microscopic details work out.' These are load-bearing admissions: without a concrete construction, the phase-space argument applies only to an invented finite-dimensional model, and the conclusion that HST models have no memory burden remains a conjecture. The manuscript should either supply a construction for the relevant unitary embeddings or state explicitly that the result is conditional on that open problem.

minor comments (4)

[Throughout] The text refers to 'Figure 1' and 'Figure 2' in Section 2, but the figures are not included in the arXiv submission; this makes the discussion of the shaded region and the non-causal-diamond geometry difficult to follow.
[Section 3, Eqs. (4) and (5)] Equations (4) and (5) are identical; one of them should be removed or replaced with the intended distinct relation.
[Section 2, model definition] The model is defined with M(M+1) Dirac fermion fields and a momentum cutoff of 'about 20 modes', but the dependence of the final conclusions on these choices, particularly on the cutoff and on the unspecified constant c in the probability e^{−c p N}, is not discussed. A brief statement about the sensitivity of the counting argument to these parameters would be helpful.
[Abstract and Section 1] The abstract notes that the conclusion depends crucially on the definition of energy and the implementation of causality; this caveat is important and should be restated prominently in the introduction, since a different choice of approximate energy would change the argument.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the toy-model phase-space argument is an internal Fermi-Golden-Rule count, not a reduction of the conclusion to its inputs.

full rationale

The paper's central claim is not derived by circular reduction. Section 2 defines an explicit finite quantum model: a Hilbert space of M(M+1) Dirac fermions, a postulated empty-diamond density matrix, and a sequence of unitary embeddings. The 'no memory burden' conclusion follows from a combinatorial count of final-state block decompositions (K!/(∏ p_i!(K−r)!)) compared with the single-block configuration (K−N ways), i.e., an internal Fermi-Golden-Rule phase-space estimate. That estimate is a computation within the model's own definitions, not an equation that is identical to an input. The model's 'energy' is indeed defined as the approximately conserved block-size sum, and the Hamiltonian is chosen to have 't Hooft-like scaling; these are model-building assumptions, not fitted parameters, and no data or external observable is 'predicted' after fitting. The self-citations to HST papers (Banks-Fischler, Banks-Zurek) supply motivation and a possible embedding, but Section 2 explicitly says the toy model 'stands on its own' and the paper flags the construction of the modular-inclusion unitaries as 'the biggest lacuna in the HST formalism'; that acknowledgment makes the conditional nature of the HST connection transparent rather than hidden. The known weakness in the Section 2 argument is that Fermi's Golden Rule requires matrix elements |V|^2, which the paper never estimates, only the density of states; but this is an incompleteness/correctness risk, not a circularity, because the conclusion is not identified with an input by construction. No fitted input is renamed as a prediction, and no self-citation is used as the sole justification for the central phase-space counting. Verdict: no significant circularity (score 0).

Assumptions & free parameters 4 free parameters · 8 assumptions · 1 invented entities

The central claim leans on a large body of the author's prior HST work, on the ad hoc choice of the toy model's dynamics, and on the unproven completion of the unitary embedding sequence. The reader is asked to take the HST framework as given, plus accept several order-one constants and cutoffs chosen by hand.

free parameters (4)

c = ~1 (order one)
The exponential suppression factor e^{-c p N} for finding a block-diagonalized state in the equilibrium density matrix; the order-one constant c is not derived.
momentum cutoff = about 20 modes
The cutoff at which Cardy's asymptotic spectral density formula begins to be a good approximation; chosen by hand for the toy model.
M in M(M+1) fermion fields = large, unspecified
The total size of the fermion system; a parameter of the toy model, not derived from data.
initial time T = large, with T/L_P taken to infinity
The finite initial time in the past; scattering states and the definition of energy depend on T, with T/L_P taken to infinity in the asymptotic argument.

assumptions (8)

standard math Fermi's Golden Rule and the principle of detailed balance govern decay rates in the large Hilbert space.
Used to argue that the phase space factor dominates decay of the meta-stable state (Section 2).
standard math Cardy's formula gives the asymptotic density of states of the U(N) current algebra.
Used to justify the cutoff at 20 modes and the spectral properties of the empty diamond density matrix (Section 2).
domain assumption The covariant entropy bound requires a UV cutoff on Dirac eigenvalues on the compact manifold and on S^{d-2}.
Used in Section 3 to motivate cutting off the Dirac spectrum on S^{d-2}.
ad hoc to paper Time evolution in a subsystem of size M has a natural time scale proportional to M ('t Hooft scaling).
Stated as 'the generalization of 't Hooft scaling to tensor models makes it easy to implement this rule' in Section 3; it is essential for the phase-space growth argument.
ad hoc to paper The empty diamond density matrix has the current-current form of Eq. (1).
Postulated as the analog of the QFT vacuum; the entire analysis relies on this ansatz.
ad hoc to paper The time-dependent Hamiltonian that performs modular embeddings is single trace.
Stated as 'it is consistent to assume that...' in Section 2; the conservation of the 'energy' depends on this.
domain assumption Bulk QFT cannot account for black hole entropy; the Carlip-Solodukhin CFT ansatz is correct.
Invoked in the Conclusions to reject the entropy model used by Dvali et al.; used to motivate the HST description.
ad hoc to paper The Hilbert bundle with the Quantum Principle of Relativity restores Poincare covariance.
Introduced in Section 3 as a proposal to connect the toy model to spacetime; the author concedes it does not show the need for higher dimensions and is not proven.

invented entities (1)

Hilbert bundle over the space of time-like geodesics with the Quantum Principle of Relativity connection
purpose: To restore Poincare covariance to the nested causal diamond construction
Proposed in Section 3 as the way to sew together descriptions along different geodesics; no independent falsifiable prediction is extracted.

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Cite this review

Pith. "Pith review of Why There is No Memory Burden in Holographic Space-time Models of Black Hole Formation and Evaporation." pith.science (2026). https://pith.science/paper/YZEF4ZZT

@misc{pith2026260812139,
  author       = {Pith},
  title        = {Pith review of: Why There is No Memory Burden in Holographic Space-time Models of Black Hole Formation and Evaporation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YZEF4ZZT}},
  note         = {Machine review of arXiv:2608.12139}
}

read the original abstract

Recent papers\cite{dvali} have claimed that general quantum information considerations put macroscopic constraints on the properties of large black holes, which can affect their lifetimes in cosmologically and astrophysically interesting ways. We examine black hole evaporation in Holographic Space Time (HST) models\cite{hstbh}, which appear to have the {\it memory burden} effect responsible for these deviations from semi-classical expectations, and explain why, in those models, no such effect exists. It is basically a consequence of Fermi's Golden Rule/The Principle of Detailed Balance, but depends crucially on both the definition of energy in HST models and the way in which causality is implemented.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

Black Hole Memory Burden and its Signatures in Gravitational Waves from Mergers

G. Dvali, M. Zantedeschi and S. Zell, “Black Hole Memory Burden and its Signatures in Gravitational Waves from Mergers,” [arXiv:2607.03560 [gr-qc]]; G. Dvali, “Swift Memory Burden in Merging Black Holes: how information load affects black hole’s classical dy- namics,” [arXiv:2509.22540 [hep-th]]; G. Contri, G. Dvali and O. Sakhelashvili, “Similari- ties i...

work page Pith review arXiv 2025

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T. Banks and W. Fischler, “Holographic space-time, Newton’s law, and the dynamics of hori- zons,” Adv. Theor. Math. Phys.27, no.1, 65-86 (2023) doi:10.4310/ATMP.2023.v27.n1.a3 [arXiv:2003.03637 [hep-th]]; T. Banks and W. Fischler, “Holographic Space-time, Newton’s Law and the Dynamics of Black Holes,” [arXiv:1606.01267 [hep-th]]; T. Banks and W. Fis- chle...

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work page

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S. Carlip, “Black hole entropy from conformal field theory in any dimension,” Phys. Rev. Lett.82, 2828-2831 (1999) doi:10.1103/PhysRevLett.82.2828 [arXiv:hep-th/9812013 [hep-th]]; S. N. Solodukhin, “Conformal description of horizon’s states,” Phys. Lett. B454, 213-222 (1999) doi:10.1016/S0370-2693(99)00398-6 [arXiv:hep-th/9812056 [hep-th]]

arXiv 1999

[5] [5]

Conformal description of near-horizon vacuum states,

T. Banks and K. M. Zurek, “Conformal description of near-horizon vacuum states,” Phys. Rev. D104, no.12, 126026 (2021) doi:10.1103/PhysRevD.104.126026 [arXiv:2108.04806 [hep-th]]

arXiv 2021

[6] [6]

Microscopic quantum mechanics of the p = rho universe,

T. Banks, W. Fischler and L. Mannelli,“Microscopic quantum mechanics of the p = rho universe,” Phys. Rev. D71, 123514 (2005) doi:10.1103/PhysRevD.71.123514 [arXiv:hep- th/0408076 [hep-th]]

work page arXiv 2005

[7] [7]

Hilbert Bundles and Holographic Space-time: the Hydrodynamic Approach to Gravity,

T. Banks, “Hilbert Bundles and Holographic Space-time: the Hydrodynamic Approach to Gravity,” [arXiv:2502.04924 [hep-th]]

arXiv

[8] [8]

CONFORMAL SUPERGRAV- ITY, TWISTORS AND THE SUPER BMS GROUP,

M. A. Awada, G. W. Gibbons and W. T. Shaw, “CONFORMAL SUPERGRAV- ITY, TWISTORS AND THE SUPER BMS GROUP,” Annals Phys.171, 52 (1986) doi:10.1016/S0003-4916(86)80023-9

doi:10.1016/s0003-4916(86)80023-9 1986